Reading 8

A hypothesis is a statement about the value of a population parameter, developed to test a theory or belief. Hypotheses are stated in terms of the population parameter being tested (e.g., the population mean \(\mu\)).

Hypothesis testing procedures use sample statistics and probability theory to decide whether a hypothesis is reasonable (not rejected) or unreasonable (rejected).

The hypothesis-testing process

1. State the hypotheses.
2. Select the appropriate test statistic.
3. Specify the level of significance.
4. State the decision rule.
5. Collect the sample and calculate the sample statistic.
6. Make a decision regarding the hypothesis.
7. Make a decision based on the results of the test.

Null and alternative hypotheses

• Null hypothesis (\(H_0\)) — what the researcher wants to reject. It is the hypothesis actually tested and is the basis for selecting the test statistic. The null always includes the "equal to" condition.
• Alternative hypothesis (\(H_a\)) — what is concluded if there is sufficient evidence to reject the null. Usually what the researcher is really trying to assess. Statistics can never prove anything; when \(H_0\) is discredited, the implication is that \(H_a\) is valid.

For \(H_0: \mu = \mu_0\), the alternative is \(H_a: \mu \ne \mu_0\). The two hypotheses are mutually exclusive and exhaustive.

Decision rule (two-tailed z-test at α = 0.05)

At α = 0.05, the test statistic is compared with critical z-values of ±1.96 (corresponding to \(\pm z_{\alpha/2} = \pm z_{0.025}\), the range within which 95% of probability lies).

\[\text{Reject } H_0 \text{ if test statistic} < -1.96 \text{ or test statistic} > 1.96\]

Each tail of the distribution beyond ±1.96 contains 0.05 / 2 = 0.025 probability.

Commonly used critical z-values

The test statistic

\[\text{test statistic} = \frac{\text{sample statistic} - \text{hypothesized value}}{\text{standard error of the sample statistic}}\]

For the sample mean \(\bar{x}\), the standard error is:

\[\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \quad \text{(when population } \sigma \text{ is known)}\] \[s_{\bar{x}} = \frac{s}{\sqrt{n}} \quad \text{(when population } \sigma \text{ is unknown; use sample } s\text{)}\]

Test statistics can follow one of four distributions: t, z (standard normal), chi-square, and F. The critical value depends on the distribution.

Type I and Type II errors

• Significance level \(\alpha\) = P(Type I error) = probability of rejecting a true null.
• Power of a test = P(correctly rejecting a false null) = 1 − P(Type II error).

Relationships and trade-offs

• Decreasing α (e.g., 5% → 1%) increases the probability of Type II error and reduces the power.
• For a given sample size, increasing power increases the probability of Type I error.
• For a given α, the only way to decrease P(Type II error) and increase power is to increase the sample size.

We reject or fail to reject the null — never "accept" it; a null can only be supported or rejected.

p-value

The p-value is the probability of obtaining a test statistic that would lead to rejecting the null, assuming the null is true. It is the smallest significance level at which the null can be rejected.

Test 1 — Value of a Population Mean

Test 2 — Difference Between Means (Independent Samples)

Used when two samples are independent and both populations are normally distributed. With unknown variances assumed equal, the pooled-variance t-test is used:

\[t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\dfrac{s_p^2}{n_1} + \dfrac{s_p^2}{n_2}}}\]

where the pooled variance is:

\[s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}\]

Degrees of freedom: \(n_1 + n_2 - 2\).

Intuition: if sample means are close, the t-statistic is small and we do not reject equality; if they are far apart, the t-statistic is large and we reject.

Test 3 — Paired Comparisons (Means of Dependent Samples)

Used when samples are not independent — observations both depend on a common factor (e.g., market returns, industry conditions, before/after the same event).

Hypotheses (two-tailed, general form):

\[H_0: \mu_d = \mu_{dz} \quad \text{versus} \quad H_a: \mu_d \ne \mu_{dz}\]

where \(\mu_{dz}\) is the hypothesized mean difference (commonly zero).

Test statistic with \(n - 1\) degrees of freedom:

\[t = \frac{\bar{d} - \mu_{dz}}{s_{\bar{d}}}\]

where:

\[\bar{d} = \frac{1}{n}\sum_{i=1}^{n} d_i, \qquad s_{\bar{d}} = \frac{s_d}{\sqrt{n}}, \qquad s_d = \sqrt{\frac{\sum_{i=1}^{n} (d_i - \bar{d})^2}{n-1}}\]

Test 4 — Value of a Population Variance (chi-square test)

Used for hypothesis tests on the variance of a normally distributed population.

Hypotheses (two-tailed):

\[H_0: \sigma^2 = \sigma_0^2 \quad \text{versus} \quad H_a: \sigma^2 \ne \sigma_0^2\]

Test statistic with \(n - 1\) degrees of freedom:

\[\chi_{n-1}^2 = \frac{(n - 1) s^2}{\sigma_0^2}\]

The chi-square distribution is asymmetric and bounded below by zero — values cannot be negative. It approaches normal as df increases.

For a two-tailed test at α = 0.05 with df = 30: critical values are 16.791 (lower) and 46.979 (upper).

Test 5 — Comparing Two Population Variances (F-test)

Used for the equality of variances of two normal populations from independent samples.

Hypotheses (two-tailed):

\[H_0: \sigma_1^2 = \sigma_2^2 \quad \text{versus} \quad H_a: \sigma_1^2 \ne \sigma_2^2\]

Test statistic (always put the larger variance in the numerator):

\[F = \frac{s_1^2}{s_2^2}\]

Degrees of freedom: \(n_1 - 1\) (numerator) and \(n_2 - 1\) (denominator).

Properties of the F-distribution

• Right-skewed, bounded below by zero.
• When sample variances are equal, F = 1.
• Lower critical value is the reciprocal of the upper critical value.
• By putting the larger variance in the numerator, we only need to consider the upper-tail critical value.

Parametric tests rely on assumptions about the distribution of the population and are specific to population parameters. The z-test, for example, requires a defined mean and standard deviation, and either a large sample (via the central limit theorem) or a normally distributed population.

Nonparametric tests either don't consider a particular population parameter, or make few assumptions about the population.

When nonparametric tests are appropriate

1. Distributional assumptions fail — e.g., testing a mean for a small sample from a non-normal distribution, where neither the t-test nor z-test is appropriate.
2. Data are ranks (ordinal measurement scale) rather than values.
3. The hypothesis doesn't concern distribution parameters — e.g., testing whether a variable is normally distributed, or whether a sequence is random. The runs test estimates the probability that a series of changes (+, +, −, −, +, −, …) is random.